Deformation analysis by an improved similarity transformation

Abstract

In this contribution, deformation analysis is rigorously performed by a non-linear 3-D similarity transformation. In contrast to traditional methods based on linear least-squares (LLS), here we solve a non-linear problem without any linearization. To achieve this goal, a new weighted total least-squares (WTLS) approach with general dispersion matrix is implemented to deformation analysis problem. Although some researchers have been trying to solve deformation analysis using TLS approaches, these attempts require modification since they used to apply unstructured TLS techniques such as Generalized TLS (GTLS) to similarity transformation which requires structured TLS (STLS) techniques while the WTLS approach preserves the structure of the functional model when based on the perfect description of the variance-covariance matrix. As a secondary scope, here it is analytically proved that LLS is not identical to nonlinear estimations such as the WTLS methods and rigorous nonlinear least-square (RNLS) as opposed to what in some contributions has been claimed. The third attainment of this contribution is proposing another algorithm for rigorous similarity transformation with arbitrary rotational angles. It is based on the RNLS method which can obtain the correct update of misclosure. Moreover, compared to transformation methods that deal with arbitrary rotational angles, we do not need to impose any orthogonality constraints here. Two case studies numerically confirm that the WTLS and RNLS methods provide the most accurate results among the LLS, GTLS, RNLS and WTLS approaches in two landslide areas.

Keywords

• Deformation analysis
• Weighted total least-squares method
• Nonlinear similarity transformation
• Structured total Least-Squares problem